Stable Phase-locking Research Articles

On the stability & phase locking to a system reference of an optoelectronic oscillator with large delay

Delay line oscillators based on photonic components, offer the potential for realization of phase noise levels up to 3 orders of magnitude lower than achievable by conventional microwave sources. Fibreoptic-based delay lines can realize the large delay required for low phase noise systems whilst simultaneously achieving insertion loss levels that can be compensated with available microwave and photonic amplification technologies. Multimode operation is an artefact of the delay line oscillator and introduces modulational instability into phase-locked control loops. An optoelectronic oscillator (OEO) with large delay under proportional integral control by a phase-locked loop (PLL) is modelled, providing the first report of the location of all the infinity of poles of the PLL-OEO system function. The first experimental observation of giant phase modulated oscillation of a free OEO and spontaneous giant phase modulated oscillation of a PLL-OEO are also reported and explained respectively as a source and manifestation of modulational instability. Nevertheless, the analysis and experimental observations, including a prototype 10 GHz PLL-OEO phase noise spectral density achieving - 80dBc/Hz {text{at}} 10 Hz and - 145dBc/Hz {text{at}} 10 kHz, demonstrate that stable phase lock operation and optimum phase noise performance is achievable provided full account of the multimode nature of the OEO is taken in the phase lock analysis.

Multiharmonic Frequency-Chirped Transducers for Surface-Acoustic-Wave Optomechanics

Wide passband interdigital transducers are employed to establish a stable phase-lock between a train of laser pulses emitted by a mode-locked laser and a surface acoustic wave generated electrically by the transducer. The transducer design is based on a multi-harmonic split-finger architecture for the excitation of a fundamental surface acoustic wave and a discrete number of its overtones. Simply by introducing a variation of the transducer's periodicity $p$, a frequency chirp is added. This combination results in wide frequency bands for each harmonic. The transducer's conversion efficiency from the electrical to the acoustic domain was characterized optomechanically using single quantum dots acting as nanoscale pressure sensors. The ability to generate surface acoustic waves over a wide band of frequencies enables advanced acousto-optic spectroscopy using mode-locked lasers with fixed repetition rate. Stable phase-locking between the electrically generated acoustic wave and the train of laser pulses was confirmed by performing stroboscopic spectroscopy on a single quantum dot at a frequency of 320 MHz. Finally, the dynamic spectral modulation of the quantum dot was directly monitored in the time domain combining stable phase-locked optical excitation and time-correlated single photon counting. The demonstrated scheme will be particularly useful for the experimental implementation of surface acoustic wave-driven quantum gates of optically addressable qubits or collective quantum states or for multi-component Fourier synthesis of tailored nanomechanical waveforms.

Variety of Alternative Stable Phase-Locking in Networks of Electrically Coupled Relaxation Oscillators

We studied the dynamics of a large-scale model network comprised of oscillating electrically coupled neurons. Cells are modeled as relaxation oscillators with short duty cycle, so they can be considered either as models of pacemaker cells, spiking cells with fast regenerative and slow recovery variables or firing rate models of excitatory cells with synaptic depression or cellular adaptation.It was already shown that electrically coupled relaxation oscillators exhibit not only synchrony but also anti-phase behavior if electrical coupling is weak. We show that a much wider spectrum of spatiotemporal patterns of activity can emerge in a network of electrically coupled cells as a result of switching from synchrony, produced by short external signals of different spatial profiles. The variety of patterns increases with decreasing rate of neuronal firing (or duty cycle) and with decreasing strength of electrical coupling. We study also the effect of network topology - from all-to-all – to pure ring connectivity, where only the closest neighbors are coupled.We show that the ring topology promotes anti-phase behavior as compared to all-to-all coupling. It also gives rise to a hierarchical organization of activity: during each of the main phases of a given pattern cells fire in a particular sequence determined by the local connectivity. We have analyzed the behavior of the network using geometric phase plane methods and we give heuristic explanations of our findings.Our results show that complex spatiotemporal activity patterns can emerge due to the action of stochastic or sensory stimuli in neural networks without chemical synapses, where each cell is equally coupled to others via gap junctions. This suggests that in developing nervous systems where only electrical coupling is present such a mechanism can lead to the establishment of proto-networks generating premature multiphase oscillations whereas the subsequent emergence of chemical synapses would later stabilize generated patterns.

Perturbations can distinguish underlying dynamics in phase-locked two-neuron networks

Synchronization of neuronal activity is observed in high- and low-level functions of the nervous system. For example, during memory tasks, neural activity in different brain regions phase-locks [1,2], while synchronized cells in the brainstem contribute to respiratory function [3]. In a two-cell system, network phase is a measure of when one neuron spikes with respect to the other neuron, normalized by the network period; phase-locked systems have a constant network phase, and here we define synchronized systems as those with a network phase close to zero. Understanding how and under what conditions neurons synchronize and phase-lock is important for understanding how neuronal populations function. Phase resetting curves (PRCs) describe how a neuron's period changes in response to inputs applied at various times during the interspike interval [4,5]. We use a PRC-based map of stimulus times vs response times [6] to predict if two coupled neurons will phase-lock, as well as how robust this phase-locking is against perturbations; two curves, one per neuron, are plotted against each other on this map, and intersections of the curves correspond to fixed points of the coupled system. Stable fixed points indicate stable phase-locking, while unstable points predict movement around the map. Close, but non-intersecting, curves result in networks that show a preferred phase with some phase slips. The fixed points (or lack thereof) on this map determine the dynamics of the coupled system, but similar statistics of network phases can be obtained from different underlying dynamics. Our goal is to discern underlying dynamic properties of the coupled system when a PRC cannot be measured. Here we explore perturbation-based methods to distinguish between different fixed point cases that result in similar network phase histograms. Because fixed points determine the network's response to perturbations, we use perturbations to uncover if, and where, fixed points exist without actually creating the PRC-based map. Parameters for coupled, conductance-based neuron models are chosen such that different numbers of fixed points produce phase histograms with similar network phases. When a synaptic perturbation or random noise is applied to one simulated neuron, the resulting trajectory of subsequent cycles around the map plane gives clues to the location and presence of underlying fixed points; different trajectories that point to the same location indicate a stable fixed point, while trajectories that point away from a region indicate an unstable point. Trajectories that only traverse the map plane in one direction are indicative of zero fixed points but close curves. Because biological neurons are not perfectly regular oscillators, we also investigate the effects of noise on the network phase histogram and how it affects our ability to resolve fixed point cases. The simulation results presented here will be validated in experimental hybrid circuits of one Aplysia californica neuron and one computational neuron coupled using the dynamic clamp.

SYNCHRONIZATION AND BIFURCATION IN LIMIT CYCLE OSCILLATORS WITH DELAYED COUPLINGS

In this paper, a system of three globally coupled limit cycle oscillators with a linear time-delayed coupling are investigated. Considering the delay as a parameter, we also study the effect of time delay on the dynamics. Next, Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction are obtained. Based on the symmetric Hopf bifurcation theorem, we investigate stable phase-locking and unstable waves. Then later, the directions of Hopf bifurcations are determined in some region, where stability switches may occur. The results show that the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulations are applied to verify the theoretical predictions.

Understanding the mechanisms underlying phase-locking behavior in the crayfish swimmeret system

One of the fundamental challenges in neuroscience is to understand how the intrinsic properties of neurons and the properties of neural networks combine to produce behavior. Networks that produce rhythmic motor behaviors, such as locomotion, provide important model systems to address this problem. A particularly good model for this purpose is the neural circuit underlying the coordinated rhythmic limb movements in the crayfish swimmeret system. During forward swimming, rhythmic movements of swimmerets on different segments of the crayfish abdomen progress from back to front with the same period, but neighboring swimmerets are phase-lagged by 25% of the period. This coordination of limb movements is maintained over a wide range of frequency. The exact mechanisms underlying this robustly stable phase-locking are not known. Here, we use mathematical modeling and analysis in conjunction with recent experimental results to obtained insight into these mechanisms. The rhythmic behavior of each swimmeret is driven by a local pattern generating circuit consisting of a half-center oscillator (HCO). These local pattern generating circuits are connected through well-described intersegmental connections. We model the neural circuitry of the swimmeret system as a chain of HCOs. First, we examine the phase response properties of HCOs for two fundamentally different mechanisms that produce anti-phase activity in HCOs: the “escape” and “release” mechanisms. We demonstrate that the “escape” and “release” mechanisms give rise to very different phase response properties, and we use phase plane arguments to explain the different shapes of the phase response curves. We then examine a chain of four nearest-neighbor coupled HCOs. We use the coupled oscillator theory and symmetry arguments to show that the phase-locking 25% phase-locking observed in the crayfish swimmeret system arises naturally from the network connectivity, but this phase-locking is robustly stable for only some combinations of connectivity and the escape/release mechanisms.

A drive-free vibratory gyroscope

Computational and analytical works have shown that certain coupling schemes can lead to significant enhancements in sensitivity, accuracy, and lower costs for a wide range of sensor devices whose output and performance depends directly on the ability of individual units to generate stable limit cycle oscillations. Vibratory gyroscopes are very good candidates for this new paradigm as their accuracy and sensitivity are directly dependent on the ability of a driving signal to produce and maintain oscillations with stable amplitude, phase, and frequency. To achieve higher accuracy, we show proof of concept of a novel scheme: a drive-free coupled gyroscope system in which the coupling alone can lead to self-regulated limit cycle oscillations in the drive- and sense-axes with stable constant amplitude and phase-locking.

Spike-Timing Dependent Plasticity and Feed-Forward Input Oscillations Produce Precise and Invariant Spike Phase-Locking

In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and the spiking of single neurons can be highly precise, across neuronal populations and cell types. Spike phase (i.e., the spike time with respect to a reference oscillation) is known to carry reliable information, both with phase-locking behavior and with more complex phase relationships, such as phase precession. How this precision is achieved by neuronal populations, whose membrane properties and total input may be quite heterogeneous, is nevertheless unknown. In this note, we investigate a simple mechanism for learning precise LFP-to-spike coupling in feed-forward networks – the reliable, periodic modulation of presynaptic firing rates during oscillations, coupled with spike-timing dependent plasticity. When oscillations are within the biological range (2–150 Hz), firing rates of the inputs change on a timescale highly relevant to spike-timing dependent plasticity (STDP). Through analytic and computational methods, we find points of stable phase-locking for a neuron with plastic input synapses. These points correspond to precise phase-locking behavior in the feed-forward network. The location of these points depends on the oscillation frequency of the inputs, the STDP time constants, and the balance of potentiation and de-potentiation in the STDP rule. For a given input oscillation, the balance of potentiation and de-potentiation in the STDP rule is the critical parameter that determines the phase at which an output neuron will learn to spike. These findings are robust to changes in intrinsic post-synaptic properties. Finally, we discuss implications of this mechanism for stable learning of spike-timing in the hippocampus.

Phase-locking of the beat signal of two distributed-feedback diode lasers to oscillators working in the MHz to THz range

We present difference-frequency stabilization of free-running distributed-feedback (DFB) diode lasers, maintaining a stable phase-lock to a local oscillator (LO) signal. The technique has been applied to coherent hybrid THz imaging which employs a high-power electronic radiation source emitting at 0.62 THz and electro-optic detectors. The THz radiation of the narrow-band emitter is mixed with the difference frequency of the DFB diode laser pair. The resulting intermediate frequency is phase-locked to the LO signal from a radio-frequency generator using a fast laser-current control loop. The stabilization scheme can be adapted readily to a wide range of applications which require stabilized laser beat-notes.

Stable phase-locking of an external-cavity diode laser subjected to external optical injection

We analyse the locking phenomena arising when an external-cavity diode laser is subjected to optical injection from another uncontrolled diode laser. The system stability is investigated as a function of coupled cavity time delay and the optical injection strength. Different regimes, spanning from ‘in-phase locking’ to ‘out-of-phase locking’ with ultimate amplitude death of low-frequency fluctuations/pulsations, are described experimentally as well as numerically for weak to moderate injection. Qualitative agreements between numerically and experimentally observed results for amplitude quenching are shown. Numerical studies describe the shifting of phase-flip bifurcation as the optical injection strength is varied for a particular time delay. Stable phase-locking behaviours, which are desired from the point of view of practical applications, are observed numerically in a wide range of control parameter space.

Dynamic synchronization and chaos in an associative neural network with multiple active memories.

Associative memory dynamics in neural networks are generally based on attractors. Retrieval based on fixed-point attractors works if only one memory pattern is retrieved at the time, but cannot enable the simultaneous retrieval of more than one pattern. Stable phase-locking of periodic oscillations or limit cycle attractors leads to incorrect feature bindings if the simultaneously retrieved patterns share some of their features. We investigate retrieval dynamics of multiple active patterns in a network of chaotic model neurons. Several memory patterns are kept simultaneously active and separated from each other by a dynamic itinerant synchronization between neurons. Neurons representing shared features alternate their synchronization between patterns, thus multiplexing their binding relationships. Our model includes a mechanism for self-organized readout or decoding of memory pattern coherence in terms of short-term potentiation and short-term depression of synaptic weights.

Forced synchronization in a system with unstable cycle

The dynamics of a system with unstable limiting cycle under the action of a periodic sequence of delta pulses is considered. It is shown that stable quasiperiodic regimes and phase lock regimes (synchronization) can be observed within a narrow range of parameters of the external signal.

Subharmonic resonance and chaos in forced excitable systems.

Forced excitable systems arise in a number of biological and physiological applications and have been studied analytically and computationally by numerous authors. Existence and stability of harmonic and subharmonic solutions of a forced piecewise-linear Fitzhugh-Nagumo-like system were studied in Othmer ad Watanabe (1994) and in Xie et al. (1996). The results of those papers were for small and moderate amplitude forcing. In this paper we study the existence of subharmonic solutions of this system under large-amplitude forcing. As in the case of intermediate-amplitude forcing, bistability between 1 : 1 and 2 : 1 solutions is possible for some parameters. In the case of large-amplitude forcing, bistability between 2 : 2 and 2 : 1 solutions, which does not occur in the case of intermediate-amplitude forcing, is also possible for some parameters. We identify several new canonical return maps for a singular system, and we show that chaotic dynamics can occur in some regions of parameter space. We also prove that there is a direct transition from 2 : 2 phase-locking to chaos after the first period-doubling bifurcation, rather than via the infinite sequence of period doublings seen in a smooth quadratic interval map. Coexistence of chaotic dynamics and stable phase-locking can also occur.

Synchronization and stable phase-locking in a network of neurons with memory

We consider a network of three identical neurons whose dynamics is governed by the Hopfield's model with delay to account for the finite switching speed of amplifiers (neurons). We show that in a certain region of the space of (α, β), where α and β are the normalized parameters measuring, respectively, the synaptic strength of self-connection and neighbourhood-interaction, each solution of the network is convergent to the set of synchronous states in the phase space, and this synchronization is independent of the size of the delay. We also obtain a surface, as the graph of a continuous function of τ = τ( α, β) (the normalized delay) in some region of (α, β), where Hopf bifurcation of periodic solutions takes place. We describe a continuous curve on such a surface where the system undergoes mode-interaction and we describe the change of patterns from stable synchronous periodic solutions to the coexistence of two stable phase-locked oscillations and several unstable mirror-reflecting waves and standing waves.

Phase-locking in weakly heterogeneous neuronal networks

We examine analytically the existence and stability of phase-locked states in a weakly heterogeneous neuronal network. We consider a model of N neurons with all-to-all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons. We consider both inhibitory and excitatory coupling. We derive the conditions under which stable phase-locking is possible. In homogeneous networks, many different periodic phase-locked states are possible. Their stability depends on the dynamics of the neuron and the coupling. For weak heterogeneity, the phase-locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable. For enough heterogeneity, phase-locked solutions either lose stability or are destroyed completely. We analyze the possible states the network can take when phase-locking is broken.

Optical phase-sensitive amplificationusing two phase-locked light sources

Optical phase-sensitive amplification using individual pump and signal light sources is presented for the first time. In the experiment, optical phase of pump and signal light is locked by an optical phase-locked loop (OPLL). Stable optical phase-locking is achieved and optical phase-sensitive amplification with a gain of 12 dB is successfully demonstrated using a nonlinear Sagnac interferometer.

Phasing and the Pickup of Optical Information in Cascade Juggling

Four experiments were conducted to examine the relationship between the phasing of hand movements and the pickup of optical information in cascade juggling. Three jugglers of intermediate skill juggled three balls while wearing liquid crystal (LC) glasses that opened and closed at preset intervals. The first experiment, in which the duration of the viewing window was gradually reduced to zero, revealed a preference for seeing the segment of the ball flight following the zenith in one subject; such a preference was hinted at in the other two subjects. The second experiment, in which the tachistoscopic rhythm of the glasses was perturbed, showed that, in the case of a stable phase lock, the phasing of the hand movements was adjusted to restore the visibility of the segment following the zenith when it was lost. The third experiment, however, revealed that, after practice, the jugglers did not become better attuned to the optical information contained in this segment. The fourth experiment, in which two jugg...

Phase locking laser diodes using photorefractive coupling

Several authors have reported strong frequency pulling while coupling laser diodes using phase conjugation. However, to our knowledge, no one has proven stable phase locking by interfering the laser outputs to show stable fringes. A stable phase lock has been obtained, and the geometry used is compared with those proposed previously.

Advances in Phase-Lock Demodulation

A procedure for the design of nearly optimum unconditionally stable phase-lock (PL) demodulators is given. The various noise contributions at threshold are investigated and procedures to reduce the noise due to losses of lock are discussed. Experimental results of a nearly optimum 24-telephone-channel PL demodulator equipped with a particular type of threshold noise compensator are given.